January 2019’s topic of **The Galleseum** –**Acrylic Math’s FREE monthly art newsletter**– is about the relationship between art and math, which dates back to antiquity and spans to modern times. And, this is the corresponding Blog Post.

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### Blog Post

An odd couple? Not really! The relationship between art and math dates back to antiquity and spans to modern times. In 4 BC, the Greek sculptor Polykleitos of Argos described the ideal mathematical proportions of the human body in a work titled the *Kanon*. ^{[1] [2]} During the Renaissance, Leonardo da Vinci, the Italian genius, also described the ideal mathematical proportions of the human body in a drawing titled *L’Uomo Vitruviano* ^{[3]} (See Figure 1: The Vitruvian Man by Leonardo da Vinci. Public domain. ^{[4]})

And, in modern times, Piet Mondrian, the famous Dutch painter, used simple geometric elements in his work ^{[5]} (See Figure 2: Composition No. III by Piet Mondrian. Public domain. ^{[6]}).

#### The Golden Ratio

The golden ratio is represented by the *Elements*. ^{[7]} Oftentimes, the golden ratio is displayed as the golden rectangle, whose sides are equal to 1: φ ^{[8]} (See Figure 3: Golden rectangle. Public domain. ^{[9]}).

Art cognoscenti have identified the use of the golden rectangle in design. For example, Samuel Obara of the Department of Mathematics of the University of Georgia recognized *Composition No. II* ^{[10]} (See Figure 4: Composition No. II by Piet Mondrian. Public domain. ^{[11]}).

#### Tessellations

Tessellations, from the Latin *tessella* (small square), are tilings of continuous shapes: Euclidean, organic, and three-dimensional. ^{[12]} Figure 5 (Penrose tiling. Public domain. ^{[13]}) displays an example of a tessellation.

Tesselations were used in ancient Rome and in the Islamic world, notably in the Alhambra, in Granada, Spain (See Figure 6: Tessellation, Alhambra, Seville, Spain. © 2007 Gruban. Reprinted with permission. ^{[14]}) In modern times, the renowned Dutch artist M.C. Escher use tessellations in his work. ^{[12]}

#### Fractals

Fractals, from the Latin *fractus* (broken), are detailed patterns that endlessly repeat themselves at different scales. Fractals are characterized by self-similarity and non-integer dimensions. ^{[15] [16]} Fractal geometry is rooted in the seminal works of Gottfried Leibniz, the ^{th}^{th}^{[17]} Figure 7 (Triangle fractal. © Fractal Foundation. Reprinted with permission. ^{[16]}) displays an example of a fractal.

#### Summary

Myriad examples document the relationship between art and math. The three topics briefly presented herein -the golden ratio, tessellations, and fractals- are but “small cogs in the large wheel” of art and math.

#### Endnotes

^{[1]}
*Mathematics and art*. (2018).
From Wikipedia. URL:
https://en.wikipedia.org/wiki/Mathematics_and_art

^{[2]}
*Polyclitus*. (2018). From
Encyclopedia Britannica. URL:
https://www.britannica.com/biography/Polyclitus

^{[3]} *Vitruvian Man*. (2018). From Wikipedia. URL: https://en.wikipedia.org/wiki/Vitruvian_Man

^{[4]} Da Vince, L. (2018). *Vitruvian Man*. From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:Da_Vinci_Vitruve_Luc_Viatour.jpg

^{[5]} Jaffe, H.L.C. (2018). *Piet Mondrian*. From Encyclopedia Britannica. URL: https://www.britannica.com/biography/Piet-Mondrian

^{[6]} Mondrian, P. (1929). *Composition No. III.* From The Athenaeum. URL: https://www.the-athenaeum.org/art/detail.php?ID=85852

^{[7]} Berry, B. (2017). *What is the golden ratio?* From Math Hacks. URL: https://medium.com/i-math/what-is-the-golden-ratio-d3cc17c8fefd

^{[8]} *Golden rectangle*. (2018). From Wikipedia’s. URL: https://en.wikipedia.org/wiki/Golden_rectangle

^{[9]} *Golden Rectangle*. (2018). From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:SimilarGoldenRectangles.svg

^{[10]} Obara, S. (n.d.). *Golden ratio in art and architecture*. From The University of Georgia, URL: http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html

^{[11]} Mondrian, P. (1030). *Composition No. II.* From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:Piet_Mondriaan,_1930_-_Mondrian_Composition_II_in_Red,_Blue,_and_Yellow.jpg

^{[12]} *Tessellation*. (2018). From Wikipedia. URL: https://en.wikipedia.org/wiki/Tessellation

^{[13]} *Penrose tiling*. (2009). From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:Penrose_Tiling_(P1_over_P3).svg

^{[14]} Gruban. P. (2007). *Tesselletion, Alhambra, Seville, Spain*. From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:Tassellatura_alhambra.jpg

^{[15]} *Fractal*. (2018). From Encyclopedia Britannica. URL: https://www.britannica.com/science/fractal

^{[16]} *What is a fractal?* (n.d.) From Fractal Foundation. URL: https://fractalfoundation.org/fractivities/WhatIsaFractal-1pager.pdf

^{[17]} *Fractal.* (2018). From Wikipedia.URL: https://en.wikipedia.org/wiki/Fractal